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7 frame boxes, 20" long and 12" wide (outside dimensions using 3/4 thick board (1 x material or 3/4 plywood)) will give you 1.66 which is pretty close to the golden ratio of 1.68, as far as length and width.

7 frame boxes 20 x 12 3/8 outside dimensions is 1.616
that would give you space for 7 frames in the box, which is a prime number, the days in a week number, lucky number by myth, the 7th day is sabbath number, etc.

I think you're on to something here, I'm going to build me up some 7 frame boxes to start using next year! Thanks so much for the idea!!!

I was somewhat lost on this, and figured others might be too. Heres the info from Wikipedia:

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887.[1]

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

7 frame boxes 20 x 12 3/8 outside dimensions is 1.616
that would give you space for 7 frames in the box, which is a prime number, the days in a week number, lucky number by myth, the 7th day is sabbath number, etc.

I think you're on to something here, I'm going to build me up some 7 frame boxes to start using next year! Thanks so much for the idea!!!

If you want to fully follow golden proportions, make your 7 frame equipment accommodate frames that are 11-3/4" deep. You will have to make your own custom frames, but they will individually approximate the golden ratio as well. The increased frame depth will make the 7-frame box close to the same volume as a standard deep.

Thanks Budster. This one had me wondering a bit too. Using Ray's measures of 20 X 12 3/8, the long side is 1.61 times longer than the short side , and the short side is 1.61 times shorter than the long side.

OK I guess this could be interesting for a mathematician but getting back to Greenthumb's original post and Ray's comments: Why would you considering this? I don't see an advantage of having a 7 frame chamber. Go with at least 8. This is at least somewhat standardized these days. But seven frames?
I'd try to stay with standardized sizes.

And having all sides the same would result in one huge mother of a box. 10 frame deeps are plenty heavy for me thank you very much.

7 frame boxes 20 x 12 3/8 outside dimensions is 1.616
that would give you space for 7 frames in the box, which is a prime number, the days in a week number, lucky number by myth, the 7th day is sabbath number, etc.

I think you're on to something here, I'm going to build me up some 7 frame boxes to start using next year! Thanks so much for the idea!!!

So that means the box has all sides cut to 20", with a depth of 12 3/8"???

How mclose would Jumbo Dadant supers, the deeps that is, be to what you all are talking about. They held 12 frames. So the super was pretty much square. And the deep frames were about as deep as they were long.

I ran across one of these hives in Ohio 20 some years ago. the guy only had the shallow supers. But even that was heavy enough to open and inspect.

MrGreenThumb...
No, I meant using standard langstroth deep frames, make the box 20" long just like any langstroth hive, but cut the width down to 12 3/8" to get the golden ratio as close as conveniently possible concerning length and width. I think doing the deeper frames to get the ratio correct from height vs. width would be good too, but I won't do that as I would want to use standard frames.